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MX4557 Complex Analysis

Complex Analysis
Author

F. Olukoya

Module Handbook

Students are asked to make themselves familiar with the information on key institutional policies which have been made available within MyAberdeen or on the university website.

These policies are relevant to all students and will be useful to you throughout your studies. They contain important information and address issues such as what to do if you are absent, how to raise an appeal or a complaint and how seriously the University takes your feedback.

These institutional policies should be read in conjunction with this Course Description Form, in which School-specific policies are detailed. Further information can be found on the University’s Infohub webpage or by visiting the Infohub.

Welcome to MX4557 Complex Analysis. The aim of the module is to further develop understanding of the concepts, techniques, and tools of calculus. Calculus is the mathematical study of variation. This course emphasises differential and integral calculus in one variable, and sequences and series of functions.

Intended Learning Outcomes (ILO’s)

By the end of the course, you should be able

  1. to state and illustrate the definitions of the concepts introduced in the course,
  2. to state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses,
  3. to demonstrate knowledge and understanding of proof techniques used in the course,
  4. to use the methods and results of the course to solve problems at levels similar to those seen in the course.

In particular, you should:

  • Be able to perform routine calculations with complex functions.
  • Understand the concept of analyticity and be familiar with the Cauchy-Riemann equations.
  • Know the necessary background and development of an elementary version of Cauchy’s theorem.
  • Know the important consequences of Cauchy’s theorem: Cauchy’s integral formulae, Liouville’s Theorem and Taylor Series.
  • Know about Laurent Series and the classification of isolated singularities.
  • Know the Cauchy residue theorem and some of its applications.

Logistics

Lectures

This course is taught in person in the Fraser Noble Building, FN156.
Lectures take place on Tuesdays 14:00–15:00 and Thursdays 13:00–14:00, weeks 27–34, 38-40, with the first lecture being on January 28, 2024.

Tutorials

Tutorials are on Fridays 9:00–10:00 in the Fraser Noble Building, FN156, and will start during the second week of lectures with the first tutorial being on February 4, 2024. Students should work through the current problem sheet (Table Table 1 indicates which problem sheet a tutorial covers. ) and attempt as many exercises as possible.

Timetable

Details of all sessions (lectures and tutorials) will be available on your timetable. Please make sure that you have the correct day, time and room for each session. You should check this regularly as there are occasionally changes, particularly in the first couple of weeks of the semester.

Table 1 displays a detailed schedule for the semester.

Week

Day

Chapter

Material

27

T

1

Complex numbers and functions

Th

1

Complex numbers and functions

F

28

T

2

Complex-valued functions

Th

2

Complex-valued functions

F

Tutorial 1: Problem Sheet 1

29

T

3

Topology of the complex plane

Th

3

Topology of the complex plane (Assignment)

F

Tutorial 2: Problem Sheet 2

30

T

4

Holomorphic functions

Th

4

Holomorphic functions

F

Tutorial 3: Problem Sheet 3 (Assignment due)

31

T

5

Power series

Th

5

Power series

F

Tutorial 4: Problem Sheet 4

32

T

6

Countour integrals

Th

6

Countour integrals

F

Tutorial 5: Problem Sheet 5

33

T

7

Cauchy's Theorem

Th

7

Cauchy's Theorem

F

Tutorial 6: Problem Sheet 6

34

T

Applications of Cauchy's Theorem

Th

8

Applications of Cauchy's Theorem (Assignment)

F

8

Tutorial 7: Problem Sheet 7

35-37

Spring Break (Assignment due)

38

T

9

Laurent series and singularities

Th

9

Laurent series and singularities

F

Tutorial 8: Problem Sheet 8

39

T

10

Residue Theorem

Th

10

Residue Theorem

F

Tutorial 9: Problem Sheet 9

40

T

11

Applications to integrals

Th

11

Applications to integrals

F

Tutorial 10: Problem Sheet 10

41

Exams

Table 1: Detailed Schedule

Attendance requirements

You should attend classes regularly and do the work of the course. If you fail to do this, you may be asked to discuss your reasons with the Course Organiser and possibly be reported to the Registry as an unauthorised withdrawal from the course.
https://www.abdn.ac.uk/students/academic-life/monitoring-and-student-progress.php

Private Study

In addition to lectures and tutorials, you should aim to spend at least six hours a week working on the course. During this time you should

  1. work through the example sheets,
  2. study your lecture notes and related texts,
  3. prepare in-course assignments.

Lecturers

This semester Dr. Feyisayo (Shayo) Olukoya will be the lecturer on the module. As mentioned above you can reach me by email at feyisayo.olukoya1@abdn.ac.uk; you should also feel free to arrange an in-person meeting my office is FN163 in the Fraser Noble; meeting virtually over teams is also an option.

MyAberdeen

All resources for the module (lecture notes, problem sheets, solutions e.t.c) will be made available on MyAberdeen at the appropriate time.

Occasionally, important information will be distributed to your university email account.

Assessment

Continuous Assessment

There are two Continuous Assessments and a final exam for this course.

  • Assignment 1 (15%): Released Thursday 13 February. The deadline is Friday 21 February at 3pm.
  • Assignment 2(15%): Released Thursday 20 March. The deadline is Friday 28 March at 3pm.

Assignments will be released on MyAberdeen and you should submit your solutions as a pdf file on MyAberdeen by the due date. Only in exceptional circumstances will work handed in after the due date be accepted for formal assessment.

Individual work

Your submission for Continuous Assessment must be your own individual work unless stated otherwise. Further information and a reference to the University Code of Conduct on Student Discipline can be found in the Information Booklet.

You are encouraged to discuss exercises for this course with other students, for example in small groups. You may shop around for ideas but it is expected that you possess and use your own critical faculties, to write up your own solutions. You must not copy sentences from fellow students. If you do not understand the solution of another student, do not attempt to use it as your own solution.

Formative assessment

At the end of each chapter in the notes, you will find problem sheets with questions addressing the content covered in that chapter. Although these sheets do not contribute to the continuous assessment component, you are strongly encouraged to attempt them as they are designed to consolidate your understanding and enhance your problem-solving skills. Full solutions are provided and will appear after the sheet or sheets have been covered in example classes.

Final Exam

The final exam will be during Exam Week and comprises 70% of the module mark. It is an unseen, closed-book examination. The examination will require you to state definitions, state (and possibly prove) results, and apply these to solving problems. You should be able to state every definition and result in the module unless they are marked in the lecture notes as non-examinable.

Resits

Resit In the summer there will be a resit examination consisting of a single two-hour exam paper. The CGS mark awarded for the resit will be the maximum of

  1. the mark obtained by using the resit examination result combined with the carried-forward CA mark as in the original diet and
  2. the mark based on the resit examination performance alone.

Calculators in exams

Calculators will not be allowed in the main examination or the resit examination.

Student Support

For advice on academic and non-academic issue (resonable adjutsments, financial, international, personal or health matters ) please contact Student Services.

You can find the University’s education policy for students by following this link.

If you have any problems with the course—mathematical or organisational—please contact me, or your class representative, or Mark Grant (who is in charge of undergraduate teaching).

Report a bug

If you encounter any issues with these notes or would like to leave feedback regarding lectures, tutorials, assessments, etc., please feel free to report a bug. I will review submissions periodically, so do share your thoughts.

  1. These are based on Ben Martin’s excellent set of notes↩︎